Laplacian Spectral Characterization of Some Unicyclic Graphs

Abstract

Let $W(n;q,m_{\mathrm{1}},m_{\mathrm{2}})$ be the unicyclic graph with $n$ vertices obtained by attaching two paths of lengths $m_{\mathrm{1}}$ and $m_{\mathrm{2}}$ at two adjacent vertices of cycle $C_q$. Let $U(n;q,m_{\mathrm{1}},m_{\mathrm{2}},\dots ,m_s)$ be the unicyclic graph with $n$ vertices obtained by attaching $s$ paths of lengths $m_{\mathrm{1}},m_{\mathrm{2}},\dots ,m_s$ at the same vertex of cycle $C_q$. In this paper, we prove that $W(n;q,m_{\mathrm{1}},m_{\mathrm{2}})$ and $U(n;q,m_{\mathrm{1}},m_{\mathrm{2}},\dots ,m_s)$ are determined by their Laplacian spectra when $q$ is even.